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3d $\mathcal{N}=4$ mirror symmetry with 1form symmetry
by Satoshi Nawata, Marcus Sperling, Hao Ellery Wang, Zhenghao Zhong
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Submission summary
Authors (as registered SciPost users):  Satoshi Nawata · Marcus Sperling · Zhenghao Zhong 
Submission information  

Preprint Link:  https://arxiv.org/abs/2301.02409v1 (pdf) 
Date submitted:  20230119 03:12 
Submitted by:  Sperling, Marcus 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
The study of 3d mirror symmetry has greatly enhanced our understanding of various aspects of 3d $\mathcal{N}=4$ theories. In this paper, starting with known mirror pairs of 3d $\mathcal{N}=4$ quiver gauge theories and gauging discrete subgroups of the flavour or topological symmetry, we construct new mirror pairs with nontrivial 1form symmetry. By providing explicit quiver descriptions of these theories, we thoroughly specify their symmetries (0form, 1form, and 2group) and the mirror maps between them.
Current status:
Reports on this Submission
Anonymous Report 2 on 2023314 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2301.02409v1, delivered 20230314, doi: 10.21468/SciPost.Report.6900
Strengths
1 The paper gives a thorough and explicit account of how to gauge certain finite cyclic subgroups of the global symmetry groups of 3d N=4 quiver gauge theories, and identify the 3d mirror theories.
2 The paper gives a large number of original results, and provides extensive computations to back the claimed dualities.
3 The scope is wide, with applications to 5d theories through the concept of magnetic quiver.
Weaknesses
1 The notion of 2group structure for the symmetry after gauging is alluded to, but not discussed explicitly. Necessary conditions for the existence of nontrivial 2group symmetries are given, but it is not clear whether they are sufficient or not.
2 The scope of the paper is not clearly defined: the authors announce in the Introduction they intend to gauge discrete subgroups $\Gamma^{(0)}$ of the flavor symmetry. It should be made clear which subgroups are considered (they seem to restrict to finite, cyclic subgroups contained in specific Cartan subgroups).
3 The paper is sometimes difficult to read, the computations being spread over the main text, Appendix B, Appendix C and Appendix D.
Report
The paper gives a thorough and explicit account of how to gauge certain finite cyclic subgroups of the global symmetry groups of 3d N=4 quiver gauge theories, and identify the 3d mirror theories. The choice of studied theories covers a wide spectrum (unitary quivers, orthosymplectic quivers, nonsimply laced quivers) and allows to connect to theories in other dimensions. In spite of the few weaknesses mentioned above, it is wellwritten and certainly deserves publication in SciPost after minor improvements are made.
Requested changes
1 The types of groups $\Gamma^{(0)}$ considered in the paper should be made explicit. Is it significantly more difficult to gauge subgroups that involve several Cartan factors (both for unitary and orthosymplectic quivers)? Also in Section 2.3, what would happen if neither $qk$ nor $kq$?
2 The discussion of 2groups and lines should be clarified, in particular whether the given conditions are necessary / sufficient. For instance in the "Comments on lines" paragraph on page 5, is the condition $\mathrm{gcd}(q,N)>1$ assumed?
3 In Section 2.1.2, it would be helpful to see an explicit example of Higgs branch Hilbert series computation, as it is the first time the mirror maps are used.
4 In Example 2. page 15, why are the descriptions expected to be equivalent? The discussion after (2.26) is unclear.
5 The notations for quivers with arrows should be made clearer, distinguishing multiplicity from charge in (2.14) and similar equations (e.g. in one additional entry to Table 3).
6 Figure 2 and Figure 7 are manifestly symmetric on the right hand side under a $\mathbb{Z}_2$ flip. Can you comment on why the left hand side is not?
The following are very minor corrections.
6 The caption of Figure 3 and (2.19) seem to have opposite conventions.
7 Below (A.1), $N$ should be $n$.
8 Above (2.6), "Cartam".
9 Brackets are missing in (B.50)
10 Above (D.105) a letter is missing.
11 p. 22 remove "=" in =even, =odd.
Anonymous Report 1 on 2023219 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2301.02409v1, delivered 20230219, doi: 10.21468/SciPost.Report.6768
Report
This article focused on theories that arise from gauging a discrete ordinary (zeroform) global symmetry of 3d N=4 gauge theories. This leads to a discrete oneform symmetry in the resulting theories. The authors proposed the mirror theory of the latter. The Higgs branch and Coulomb branch Hilbert series of the mirror pairs were computed. They serve as a test of the proposals and a tool to study the global symmetry of the problem. Under certain appropriate conditions, the oneform symmetry may form an extension with the flavor symmetry into a socalled twogroup symmetry. In such cases, the authors also mapped it across the mirror theories. The results in this article are interesting and could be useful for future reference. The referee recommends it for publication in the SciPost after a minor improvement.
Requested changes
The referee understood that the authors have used the arrows in Eqs. (2.8) and (2.20) to denote chiral multiplets and at the same time the higher U(1) gauge charges, as explained in Table 3 in Appendix A. This could lead to a potential confusion. The referee recommends the authors to explain clearly this point in words in both places.
Author: Marcus Sperling on 20230414 [id 3587]
(in reply to Report 1 on 20230219)
We thank the referee for thoroughly reading of the manuscript.
In view of the requested changes, we have added clarifications to both eq. (2.8) and (2.20).
Author: Marcus Sperling on 20230414 [id 3588]
(in reply to Report 2 on 20230314)We are grateful to the referee for carefully reading the draft and offering valuable suggestions.
In view of the points raised, we have revised the draft as follows: 1) We have clarified the considered groups $\Gamma$ in the introduction, p. 2, 3rd paragraph, 4th sentence. Moreover, in Section 2.3, p.12, 1st paragraph we have commented that the restriction to kq and qk is for convenience. In the sense that this choice offers straightforward quiver descriptions, while more general choices lack thereof, even though these cases are welldefined. Lastly, we have included some comments on future generalisation in the “open questions” paragraph on p.37. Such as gauging a diagonal $Z_q$ in two Cartan factors.
2) The referee correctly pointed out the missing conditions for the existence of a 2group. We have added necessary and sufficient conditions in the discussion on p.5, below eq. (2.5). Since all unitary quiver type examples consider in this paper are of this type, the considerations are analogous in later sections. We have mentioned the nontriviality of the Postnikov class in several places for the convenience of the reader.
In “Comments on lines” , p.5, the condition gcd(q,N)>1 is not assumed.
3) First explicit Hilbert series are already provided in eqs. (2.1)(2.4). We have supplemented the results of Section 2.1.2 by explicit calculations in Appendix D, specifically eqs. (D.7)(D.11).
4) We have elaborated on the equivalence of (2.25) and (2.26) in the first paragraph of p.16. The Higgs branch side offers a clear perspective, as the discrete $Z_6$ action can be moved from one set of fundamental flavours to the other by a global rotation. Similarly, the Coulomb branch side allows to reach the same conclusion by an analysis of the set of balanced nodes and the position (and balance) of the overbalanced node.
5) We have added clarifications in the text below (2.14) and in similar equations.
6) We have commented on the choice of how the discrete $Z_q$ symmetry acts on the fundamentals flavours in the captions of Figure 2 and 7. The reason behind is the freedom to apply an overall U(1) global rotation. This may have been somehow hidden in Appendix C.
We have corrected the other typos pointed out by the referee.